In the last decade,
the non-uniqueness of weak solutions to various hydrodynamic models
has been proved successfully by convex integration methods.
Originating from the work by Nash [1]},
the convex integration scheme was applied in the pioneering paper
by De Lellis-Székelyhidi [2]} in 2009
to the 3D Euler equations,
where bounded solutions with compact support in space-time were obtained (see also [3]}, [4]}).
Afterwards, a remarkable progress is the resolution of
the famous Onsager conjecture for the 3D Euler equations
by Isett [5]}, and
by Buckmaster-De Lellis-Székelyhidi-Vicol [6]} for dissipative solutions.
We also refer to [7]}, [8]}, [9]} for Euler equations,
[10]}, [11]}, [12]}, [13]} for transport equations,
and [14]} for SQG equations.
See also the surveys [15]}, [16]}, [17]} on convex integration methods.
